Cross-relaxation studies with optically detected magnetic resonances innitrogen-vacancy centers in diamond in an external magnetic field

Reinis Lazda1,∗ Laima Busaite1,† Andris Berzins1, Janis Smits1, MarcisAuzinsh1, Dmitry Budker2,3,4, Ruvin Ferber1, and Florian Gahbauer1

1Laser Centre, University of Latvia2Helmholtz-Institut, GSI Helmholtzzentrum für Schwerionenforschung, 55128 Mainz, Germany

3Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany and4Department of Physics, University of California at Berkeley, USA

In this paper cross-relaxation between nitrogen-vacancy (NV) centers and substitutional nitrogenin a diamond crystal was studied. It was demonstrated that optically detected magnetic resonancesignals (ODMR) can be used to measure these signals successfully. The ODMR were detected ataxial magnetic field values around 51.2 mT in a diamond sample with a relatively high (200 ppm)nitrogen concentration. We observed transitions that involve magnetic sublevels that are split bythe hyperfine interaction. Microwaves in the frequency ranges from 1.3 GHz to 1.6 GHz (mS =0 −→ mS = −1 NV transitions) and from 4.1 to 4.6 GHz (mS = 0 −→ mS = +1 NV transitions)were used.To understand the cross-relaxation process in more detail and, as a result, reproduce measured

signals more accurately, a model was developed that describes the microwave-initiated transitionsbetween hyperfine levels of the NV center that are undergoing anti-crossing and are strongly mixed inthe applied magnetic field. Additionally, we simulated the extent to which the microwave radiationused to induce ODMR in the NV center also induced transitions in the substitutional nitrogen viacross-relaxation.The improved understanding of the NV processes in the presence of a magnetic field will be useful

for designing NV-diamond-based devices for a wide range of applications from implementation ofq-bits to hyperpolarization of large molecules to various quantum technological applications such asfield sensors.

PACS numbers: 76.30.Mi,76.70.Hb,75.10.Dg

I. INTRODUCTION

Negatively charged nitrogen-vacancy (NV) color cen-ters in diamond crystals currently are considered as ex-cellent candidates for a very broad range of applications.Ranging from q-bits implemented on NV centers [1, 2]and quantum memory elements [3] for quantum comput-ers, to probes for different physical fields like magneticfield [4, 5], electric field [6], and temperature [7, 8]. NVcenters could serve as tools for achieving hyperpolariza-tion of a variety of molecules as well [9–11]. NV centersin diamond can also be used as probes for electron spinand nuclear spin resonance measurements of spins andnanoscale magnetic structures attached to the surface ofa diamond crystal [12–19]. The recent progress in thestudies of NV centers in diamond suggests that very soonthese studies will move from the stage of quantum physicsand quantum information science to a stage where theywill be implemented in quantum technologies [20].

Inside the diamond crystals exist a wide range of para-magnetic defects. Among these defects are substitutionalnitrogen (P1 center), which occurs when one of the 12Catoms is replaced by a nitrogen atom 14N, or point-likedefects, which can occur, for example, when one of the

∗Electronic address: reinis.lazda@lu.lv†Electronic address: laima.busaite@lu.lv

carbon 12C isotopes is replaced by a carbon 13C. NVcenters can be used as efficient sensors that are alreadyinside the diamond crystal to study these defects [21–23],which are interesting not only by themselves but for therole they play in the hyperpolarization effects of the nu-clear spin of nitrogen. Initial hyperpolarization of thenitrogen nucleus through optically pumped NV electronshave a broad range of applications [24], but can be influ-enced by other point-like defects in the diamond crystallattice [25].

In the study reported in this paper we examine cross-relaxation processes – energy exchange between hyperpo-larized NV centers and substitutional nitrogen (P1 cen-ters) in a diamond lattice when the energy splitting of themagnetic sublevels in the NV center matches the mag-netic sublevel splitting in the P1 center at a particularvalue of the applied external magnetic field B. The sub-stitutional nitrogen has an electron spin S = 1/2. Inour study we use the negatively charged NV centers thathave an extra electron combined with one electron fromthe vacancy, which results in an S = 1 system calledNV−, denoted NV in this work. NV0 centers with anunpaired electron exist in diamonds as well, but will notbe considered in this study.

Changes caused by cross-relaxation in optically polar-ized NV centers can be measured by means of opticallydetected magnetic resonances (ODMR) [26]. The prop-erties of these magnetic resonances that make them es-pecially suited for cross-relaxation studies include spin-

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1 J

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polarization-dependent fluorescence intensity, the cou-pling of the electron spin to the magnetic environment,and a spin polarization lifetime that is long compared toother similar systems, even at room temperature [27].

Magnetic dipole transitions, caused by an applied mi-crowave (MW) field, between spin magnetic sublevels inNV centers can be used to monitor the electron spin po-larization of the NV centers and to study processes suchas cross-relaxation that influence the electron spin polar-ization.

The advantage of using a continuous-wave (CW)ODMR experiment in comparison to a pulsed experiment(for example, [16]) is that there is no need for synchro-nised pulse setups. Also one can use lower MW power,and MW homogeneity does not play such an importantrole in CW ODMR experiments as in the pulsed fieldexperiments.

II. CROSS-RELAXATION PROCESS

Because NV centers can be optically initialized andread out, they can be used to study other paramagneticdefects in the local environment, as well as interactionsbetween defects. Here, we investigate the dynamics ofcross-relaxation between NV and P1 centers.

There exist two families of P1 centers in the diamondcrystal depending on whether the symmetry axis of theunpaired electron’s orbital is parallel to the NV center’saxis ("on-axis"), which is also the direction of the ap-plied magnetic field, or along one of the other three pos-sible axes in the diamond crystal that are not alignedalong the external magnetic field ("off-axis") [17]. It isalso important to remember that each of the P1 centersswitches between all four symmetry axes on a time scaleof a few milliseconds [28, 29]. This time scale is muchshorter than our measurement time, and as a result wesee signals corresponding to all possible orientations si-multaneously for every P1 center. This hints us that on-axis cross-relaxation peaks could be roughly three timesweaker than the off-axis cross-relaxation peaks.

As discussed already in 1959. [30], the angular mo-mentum of a spin system is not generally conservedin cross-relaxation processes. Similar to the Barnettand Einstein-de Haas effects, the balance of the angu-lar momentum is transferred to the rigid lattice. Cross-relaxation has been experimentally observed in a varietyof spin systems [18, 31–34]. There are no strong selectionrules for electron spin S projection changes ∆mS or nu-clear spin I projection changes ∆mI in cross-relaxationprocesses. The only requirement is that there is an en-ergy matching between the energy-level splitting in thetwo sub-ensembles [18], in our case the NV centers andthe P1 centers.

In this situation the magnetic dipole–dipole interac-tion [17] couples the two transitions (one of the NV cen-ter’s states with one of the P1 center’s states) and leadsto an efficient energy exchange between these two sys-

mI mS

10

-11/2

10

-1-1/2

10

-1-1

48 49 50 51 52 53 5410

-10

Magnetic field (mT)

Leve

l ene

rgy

/ h (M

Hz)

FIG. 1: Level diagram for NV (black) and P1 (blue) centerswith hyperfine interaction in the vicinity of 51.2 mT. The ar-rows show the energy-matching transitions between the NVcenter’s |mS = 0,mI〉 and |mS = −1,mI〉 magnetic sublevels,and the P1 center’s |mS = − 1

2,mI〉 and |mS = 1

2,mI〉 mag-

netic sublevels. Each of the arrows are split into three by theNV center’s hyperfine structure, but they are omitted in thefigure for better readability.

tems (cross-relaxation process). This energy exchangeleads to an increase in the effective relaxation rate of theNV center [16, 17]. One can expect that, as a result, thecontrast of the ODMR signals will be decreased, and thewidth of these resonances will be increased. Both param-eters can be experimentally measured and these are theeffects that we are investigating in this paper.

Besides the electron spin S = 1/2 due to the nitrogenvalence electron, the P1 center has a nuclear spin I = 1due to the nitrogen nucleus. However, these spins aredecoupled by the internal interactions in the diamondcrystal and by the applied external magnetic field B.Likewise, the NV center, in addition to the electron spinS = 1, has a nuclear spin I = 1 due to the nitrogen atomof the NV center. These spins are decoupled for the samereasons as in the P1 center. At magnetic field valuesaround 51.2 mT the energy difference between the de-coupled hyperfine-structure magnetic sublevels for the P1center |mS = +1/2,mI〉 and |mS = −1/2,mI〉 coincideswith the energy difference between the decoupled hyper-fine magnetic sublevels for the NV center |mS = −1,mI〉and |mS = 0,mI〉, and cross-relaxation can occur. Asthe nuclear gyromagnetic ratio in nitrogen γ14N is muchsmaller than the electron gyromagnetic ratio γe, let usneglect the nuclear spin for the moment.

At zero magnetic field the spin projection states for theP1 center electron spin states |mS = +1/2〉 and |mS =−1/2〉 have equal energy, but the spin states for the NVcenter |mS = 0〉 and |mS = ±1〉 are split by the zero-fieldsplitting, which is approximately equal to 2.87 GHz [27],see FIG. 2. Since we neglect nuclear spin, both centersare pure electron spin states and have the same magneticmomentum, approximately equal to two Bohr magnetonsγe ≈ 2µB . With these parameters the energy splittings ofthe electron spin projection states described above have

3

equal values around the magnetic field value of 51.2 mT,see FIG. 1.

III. ODMR METHOD

The ODMR method can be described using the NVcenter energy-level scheme shown in FIG. 2. Green laserlight was used to excite optically the NV centers. Inthe model we assume that the excitation transitions arespin-projection conserving, ∆mS = 0, irrespective of thelaser light polarization. We also assumed that the ab-sorption rates for different laser polarizations and dif-ferent transitions are equal for all three spin-projectionstates, Γp in FIG. 2). After light absorption and rapidrelaxation in the phonon band, the excited-state mag-netic sublevels either can decay back to the ground statewith equal rates Γ0 and radiate light with a wavelengthin the red part of the electromagnetic radiation spectrumor undergo non-radiative transitions to the singlet level1A1 (see FIG. 2). These non-radiative transitions occurroughly five times more rapidly from the excited stateelectron spin sublevels mS = ±1 than from mS = 0. Af-ter that the singlet-singlet transition 1A1 −→ 1E takesplace with almost all the energy being transferred in anon-radiative way with a small portion being IR radia-tion. And finally, non-radiative transitions from 1E tothe triplet ground state occur with roughly equal tran-sition probabilities to all three electron spin projectioncomponents of the 3A2 level. It can be understood thatthe differences between the non-radiative transition ratesin the excited triplet state 3E lead to the situation thatafter several excitation-relaxation cycles the populationof the NV centers in the triplet ground state will be trans-ferred to the magnetic sublevel mS = 0, and the electronspin angular momentum will be strongly polarized [27].

This electron-spin polarization in the ground state ofthe NV centers leads to an increase in the red fluorescenceintensity. If in addition to the laser radiation, MW radi-ation is applied in resonance to a certain transition fromthe mS = 0 spin polarized state to one of the mS = ±1states, the population of NV centers in the mS = 0 stateis reduced, and hence, the fluorescence intensity dimin-ishes. This effect is the essence of the ODMR method.

By analyzing the dependence of the experimentallymeasured ODMR signal contrast and width on the ex-ternal magnetic field we studied NV and P1 center cross-relaxation at magnetic field values around 51.2 mT.

IV. HYPERFINE SPLITTING, HYPERFINELEVEL MIXING AND NUCLEAR SPIN

POLARIZATION

Measurements and signal analysis are complicated bythe fact that each of the P1 and NV spin magnetic sub-levels in diamond are split into three components by thehyperfine interaction of electron spin with the nuclear

FIG. 2: Level scheme of an NV center in diamond, mS isthe electron spin-projection quantum number, Dg and De arethe ground-state and excited-state zero-magnetic-field split-tings, ΩMW is the MW Rabi frequency, γg

0 and γg±1 are the

relaxation rates from the singlet state 1E to the triplet groundstate 3A2, γe

0 and γe±1 are the relaxation rates from the triplet

excited state 3E to the singlet state 1A1 [35].

spin (I = 1) of nitrogen 14N. This hyperfine interactionis substantially smaller in the NV center than in the P1center, see Eq. (1). As a result, in our experiment with adiamond sample of 200 ppm 14N concentration in cross-relaxation transitions involving the individual hyperfinecomponents of the P1 centers the signals can be resolved,but individual hyperfine components of NV centers can-not be resolved, see FIG. 1 as well as FIG. 5 and FIG. 6in section VIA.

For the analysis of the obtained results we look at twoHamiltonians. The first one describes the NV center inthe ground state and includes the zero-field splitting, thehyperfine interaction in the NV center, and the inter-action of the NV center with an external magnetic field.We find the eigenvalues and eigenfunctions to identify thetransition energies and estimate the probabilities of thesetransitions. The second Hamiltonian includes hyperfinesplitting and interaction with an external magnetic fieldfor a P1 center. The energy states of NV center and P1center are written in the uncoupled bases |mS ,mI〉.

The Hamiltonian for the NV center with the hyperfineinteraction in an external magnetic field B directed alongthe z-axis, which coincides with the NV-axis, is [17]:

HNV = DS2z +γeB·S+S·ANV ·I+QNVI

2z−γ14NB·I, (1)

4

where ANV is a diagonal tensor of the hyperfine interac-tion between the electron spin S and the nuclear spin Iin NV center,

ANV =

ANV⊥ 0 00 ANV⊥ 00 0 ANV‖

, (2)

and D = 2870 MHz is the zero-field splitting of theground-state electronic components with mS = 0 andmS = ±1. The strengths of the electron and thenuclear spin interaction with the external magneticfield are determined by the gyromagnetic ratios γe =28.025 MHz/mT and γ14N = 3.077 kHz/mT [36]. Thehyperfine magnetic dipole interaction parameters areANV‖ = −2.14 MHz, ANV⊥ = −2.70 MHz and the hy-perfine electric quadrupole interaction parameter QNV =−4.96 MHz [17].

The Hamiltonian for the substitutional nitrogen (P1)is [17]:

HP1 = γeB · S− γ14NB · I + S · A14N · I +Q14NI2z , (3)

where A14N is the diagonal hyperfine interaction tensorbetween the electron spin S and the nuclear spin I of14N. A14N has the same mathematical structure as ANV(2). The Hamiltonian can be used for on-axis P1 cen-ters as well as for off-axis P1 centers allowing for trans-verse magnetic field components Bx and By. The hy-perfine interaction parameters for this Hamiltonian (3)are A14N‖ = 113.98 MHz, A14N⊥ = 81.34 MHz, and thequadrupole interaction parameter Q14N = −3.97 MHz[17]. From the hyperfine interaction’s tensor-componentvalues we see that the magnetic dipole part of the hyper-fine interaction for the substitutional nitrogen is muchstronger than that for the NV center. This can be ex-plained by the fact that due to the spherical symmetryof the P1 center, the electrons are located close to the ni-trogen atom. For the NV center, the electrons are mostlylocated around the vacancy, so they are separated by onelattice constant from the nitrogen atom [27].

With these Hamiltonians one can calculate at whatmagnetic field strength a particular hyperfine energy-level splitting of the P1 center coincides with a particularhyperfine energy-level splitting of the NV center, whichis the condition for cross-relaxation to occur.

V. EXPERIMENTAL SETUP

The sample used in this experiment was a high-pressure, high-temperature (HPHT) diamond crystalwith a (100) surface cut. To create NV centers it wasirradiated with electrons. The irradiation dose was 1018

e− (cm−2) at an irradiation energy of 10 MeV. After ex-posure the crystal was annealed for 3 hours at 750 C.The diamond crystal had a relatively high initial concen-tration of nitrogen 14N of around 200 ppm [4].

FIG. 3: Experimental setup.

Figure 3 shows the experimental setup used, whichis similar to the setup used in one of our previous ex-periments (see [35]). Light with a wavelength of 532nm (Coherent Verdi Nd:YAG) was delivered to the sam-ple via an optical fiber with a core diameter of 400 µm(numerical aperture 0.39). The same fiber was used tocollect the red fluorescence light, which was separatedfrom the residual green reflections with a dichroic mirror(Thorlabs DMLP567R) and a long-pass filter (ThorlabsFEL0600). The sensing volume for the experiment wasabout 0.4 × 0.4 × 0.35 mm3 determined by the dimen-sions of the optical fiber and the diamond. The samplewas glued to the end of the optical fiber using an ad-hesive (Electron Microscopy Sciences CRYSTALBOND509). The signals were recorded and averaged on a digi-tal oscilloscope (Yokogawa DL6154).

To generate the low-frequency MW field (1.3 GHz to1.6 GHz, mS = 0 −→ mS = −1 transition) a functiongenerator (SRS SG386) with power amplifiers (Minicir-cuits ZHL-2-12 and ZHL-16W-43+) for different MW fre-quency ranges were used. For the high-frequency MWfield (4.1 GHz to 4.6 GHz, mS = 0 −→ mS = +1 transi-tion) the same function generator (SRS SG386) was com-bined with a different power amplifier (Minicircuits ZVE-3W-83+). The MW field was delivered using a copperwire in close proximity (at a distance of less than 1 mm)to the diamond sample, behind the diamond and the op-tical fiber.

VI. SIGNAL ANALYSIS AND RESULTS

A. Parameters used to characterize the ODMRsignals

To align the diamond crystal in the external magneticfield, we initially measured the dependence of the laserinduced fluorescence on the strength of the applied mag-netic field, similarly to the previous work [4]. Aligningwas done using the fluorescence features around 102.4 mTwhere the ground-state level anti-crossing (GSLAC) oc-curs.

5

A typical observed fluorescence signal as a function ofthe external magnetic field is depicted in FIG. 4. Narrowpeaks or collections of peaks indicate the existence ofcross-relaxation. Near 51.2 mT, cross-relaxation betweenthe NV and P1 centers is observed.

At a magnetic field of around 59.0 mT cross-relaxationbetween the on-axis and the off-axis NV centers is ob-served. The off-axis directions are along the three pos-sible axes in the diamond crystal that are not alignedwith the applied external magnetic field. But these cross-relaxation processes were not studied in detail here.

We measured ODMR signals and from these signals ob-tained the dependence of the signal-contrast and widthon the external magnetic field strength. The ODMR sig-nal contrast (FIG. 5a and FIG. 6a) was calculated as1 − min

max (see FIG. 7). We measured the position, con-trast, and width of cross-relaxation signals for two setsof transitions.

Initially we used ODMR signals in the MW frequencyrange of 1.10− 1.40 GHz where the NV center’s electronspin transitions between the ground state magnetic sub-levels |mS = 0,mI〉 and |mS = −1,mI〉 are involved,meaning that both the |mS = 0,mI〉 and the |mS =−1,mI〉 NV center’s sublevels of the ground state MWtransition are involved in the cross-relaxation processwith the P1 center’s |mS = − 1

2 ,mI〉 ⇐⇒ |mS = 12 ,mI〉

transition.Secondly, we registered the ODMR signals in the MW

frequency range of 4.10 − 4.40 GHz, which correspondsto the electron spin transitions between the magneticsublevels |mS = 0,mI〉 and |mS = +1,mI〉 in the NVcenter. In this transition only the lower ground statelevel of the NV center |mS = 0,mI〉 is directly in-volved in the cross-relaxation process with the P1 cen-ter’s |mS = − 1

2 ,mI〉 −→ |mS = 12 ,mI〉 transition, while

the higher |mS = +1,mI〉 ground state energy level ofthe NV center is unperturbed.

We observed two peculiarities in FIG. 5 and FIG. 6.First, for the NV transition |mS = 0,mI〉 −→ |mS =−1,mI〉 the average ODMR signal contrast is at least twotimes larger than that for the |mS = 0,mI〉 −→ |mS =+1,mI〉 transition. Second, if we compare changes in theODMR signal contrast and width for a specific transition,we see that the central peak at 51.2 mT (E in FIG. 5 andFIG. 6) exhibits the largest changes in contrast and widthwith the magnetic field.

The first peculiarity can be attributed to the abovementioned fact that for |mS = 0,mI〉 −→ |mS = −1,mI〉transition in cross-relaxation conditions the relaxationrate for both the initial and the final states increases,but for |mS = 0,mI〉 −→ |mS = +1,mI〉 transition onlythe initial level is broadened by the increased relaxationrate.

The second peculiarity is due to the fact that, as willbe shown in more detail in the next Section, this centralpeak corresponds not to one, but to several transitionsin P1 center, see FIG. 10.

The excited-state energy levels undergo crossing at

around 51.2 mT as well, so the transition probabilitiesfrom the excited state to the ground state change aroundthese magnetic field values due to the crossing and mix-ing of the excited state |mS = −1〉 and |mS = 0〉 energylevels. This crossing might have an effect on the nuclearspin polarization of the NV center. We assume that thiseffect is not dominant in this case since the ESLAC oc-curs over a relatively broad frequency range that extendsacross the entire range of magnetic field values used inthis experiment [27, 37]. Also, the nuclear spin polariza-tion of the NV center remains high at the magnetic fieldvalues around 51.2 mT with only a small decrease at theESLAC point [38]. Furthermore, in samples with low P1concentration, the fluorescence structure at the magneticfield that leads to ESLAC consists mainly of one peak,which indicates that the rich structure originates mostlyfrom cross-relaxation processes [4].

A detailed analysis of the cross-relaxation transitionsinvolved in the formation of the peaks detected in the sig-nals are presented in FIG. 10 and reflected in the analysisof this figure.

Cross-relaxation transitions that involve individual hy-perfine structure components of the NV centers are hid-den under the unresolved ODMR signal measured in thevicinity of the magnetic field of 51.2 mT (see FIG. 7).

Similar cross-relaxation transitions have been stud-ied before in [16, 17] using a substantially differentmethod. The authors measured the dependence of thespin-relaxation time T1 on the magnetic field strength inthe vicinity of the cross-relaxation resonances by meansof sequences of optical and MW pulses. Furthermore, intheir analysis the authors of [16, 17] neglected the hy-perfine structure in NV centers which, as will be shownbelow, can lead to an interpretation that does not de-scribe all the features of the observed cross relaxationsignals.

FIG. 4: NV Fluorescence signal dependence on the magneticfield strength.

At the start of the signal analysis it is worth notingthat in the results of our experiment and the experimentsof [16, 17] there are some qualitative differences. Peaksdenoted as D and F in our study (see FIG. 5 and FIG. 6),are associated with transitions that in the supplemen-tary information of [16] are called disallowed transitions.

6

FIG. 5: NV center transitions |mS = 0,mI〉 −→ |mS = −1,mI〉, the solid vertical lines indicate cross-relaxation resonancepositions between the on-axis P1 centers and NV centers, the dotted vertical lines indicate cross-relaxation resonance positionsbetween off-axis P1 centers and NV centers. Images a) and b) show the measured ODMR signal resonance contrast and width(respectively) that are extracted from the experimental signals.

FIG. 6: NV center transitions |mS = 0,mI〉 −→ |mS = +1,mI〉, the solid vertical lines indicate cross-relaxation resonancepositions between the on-axis P1 centers and NV centers, the dotted vertical lines indicate cross-relaxation resonance positionsbetween off-axis P1 centers and NV centers. Images a) and b) show the measured ODMR signal resonance contrast and width(respectively) that are extracted from the experimental signals.

7

FIG. 7: Typical, measured ODMR signal (normalized, arbi-trary units) from the NV centers at around 51.2 mT externalmagnetic field for the mS = 0 −→ mS = −1 transition withunresolved hyperfine structure, showing the full width at halfmaximum for the three Lorentzians and the contrast of theODMR signal.

These transitions, according to [16], correspond to simul-taneous flip-flop between the on-axis P1 electron and nu-clear spins. In [16], these peaks are noticeably weakerthan in this study.

When we take into account the nuclear spin of the NVcenter as well, a new partner appears in the analysis ofthe angular momentum projection balance. As a result,we will not be using the notion of allowed and disallowedtransitions in the present analysis.

B. Simultaneous interaction of MW with NV andP1 centers

We think that the difference in the relative intensityof the peaks is due to the fact that at the magnetic fieldstrength where cross-relaxation occurs and the energy-level splittings in P1 and NV centers are equal, the MWfield that is applied to measure ODMR signals is reso-nantly coupled not only to the hyperfine transitions of theNV center, but of the |mS = 0,mI〉 −→ |mS = −1,mI〉MW hyperfine transition of the P1 center as well.

Let us make rough estimates on how MW radiation si-multaneously coupled to the resonant transitions for bothpartners of the cross-relaxation process can influence thesignals.

To investigate the impact of the two different ap-proaches (pulsed MW vs CW ODMR experiment) we de-veloped a model that includes a P1 center dipole-coupledto a proximal NV center (nuclear contributions are ig-nored). The Hamiltonian that describes the evolution ofthis system consists of three parts:

Htot = HNV + HP1 + Hint, (4)

where HNV and HP1 are the Hamiltonians of the NV andP1 centers, respectively, as defined in equations (1) and(3) without the nuclear contributions, and Hint describesthe dipolar interactions between the two which can be

FIG. 8: Rotating-frame Hamiltonian after fast-oscillatingterms have been set to zero for the energy matched (onlythe black squares) and unmatched (black and grey squaresindicating the matrix elements related to the MW) cases.The kets surrounding the figure describe the ordering of the|mNV,mP1〉 electron-spin basis states.

written as:

Hint = K1

[SNV · SP1

r3− 3

1

r5

(SNVr

)·(SP1r

)], (5)

where K1 is a constant that describes the interactionstrength, r is the vector that connects the two point-like defects in diamond. If, in addition, the MW field isapplied, its action is described as:

HCW =Htot+

+ Ω1 cos (ωt) SNVx + Ω2 cos (ωt) SP1x,(6)

where ω is the applied MW frequency and Ω1,2 are thecorresponding Rabi frequencies. For this model calcula-tion we can assume Ω1,2 to be equal, as the g-factors ofP1 and NV centers are similar, and ω is assumed to beresonant with the corresponding spin transitions in bothNV and P1 centers. To simplify the problem it is treatedin the rotating frame [39, 40], where the effective Hamil-tonian can be obtained by the following transformation:

Hrot =ei(HP1+HNV)tHCWe−i(HP1+HNV)t−

− (HP1 + HNV).(7)

A visual representation of the obtained interaction ma-trices in the |mNV,mP1〉 electron spin basis are shown inFIG. 8.

Initially the system is assumed to be totally polarizedin the electron spin |mNV = 0〉 state while the P1 spin isassumed to be unpolarized. Then the system is allowed toevolve under the influence of the effective rotating frameHamiltonian and the P1 polarization is monitored (de-fined as the expectation value of 〈SP1z〉). Two cases, onewith no MW field and one with a field much stronger

8

0 2 4 6 8 10 12

-0.5

0.0

0.5

1.0

Time

P1polarization⟨Sz⟩

MWs off

MWs on

FIG. 9: Expectation value 〈SP1z〉 as a function of time withand without a microwave field.

than the average NV-P1 dipole coupling strength (K1)value, are shown in FIG. 9. While the peak polarizationof the P1 centers is slightly higher without the MW field,the transfer rate due to Rabi oscillations is much morerapid with the field on.

We can assume that the rapid changes in the P1 elec-tron spin polarization when the MW field is on can beinterpreted as an increase of the spin relaxation rate inthe presence of the MW field. This increased effectiverelaxation decreases the ODMR signal contrast and in-creases the signal width.

As far as the MW signal influences both partners in thecross-relaxation process we cannot expect that the rela-tive peak intensities in pulsed experiments [16, 17] andthis ODMR experiment will be the same. The reason isthe following: in a pulsed experiment, after the initial po-larization of the NV centers, they are left to relax duringa time of field-free interaction with the P1 centers. Andafter a certain interaction time the laser field is switchedon for readout. On the contrary, in our experiment, theoptical and the MW fields are on all of the time. Andfinally, most probably the introduction of the neglectednuclear spin, which is not directly coupled to the MWradiation into the model, will not qualitatively alter theconclusions derived from this simple model.

Let us now come back to the analysis of the influenceof the NV nuclear spin on the cross-relaxation signals.It is seen experimentally and understood in the modelsimulations that, due to hyperfine mixing in the externalmagnetic field, most of the population of the NV centeris polarized to the |mS = 0,mI = +1〉 state, which doesnot mix with other states [38, 41].

At the temperatures used in the experiment the P1 en-ergy levels are populated almost equally. The relative dif-ference of the populations is around 2 · 10−4, determinedby the Boltzmann distribution, with the |mS = − 1

2 〉 en-ergy level being slightly more populated (due to the factthat it has a lower energy in the applied external mag-netic field).

C. Detailed analysis of the differentcross–relaxation transitions

The ODMR signal contrasts depicted in FIG. 5a andFIG. 6a are obtained from the ground-state unresolvedhyperfine MW transitions (see FIG. 7). The magneticfield value at each point is determined from the ODMRpeak position (the mean position value of the three hy-perfine components that make up the unresolved ODMRspectral peak). The full width at half maximum ofthe individual assumed Lorentzian-shaped componentswas determined by fitting the data with a sum of threeLorentzian functions with equal widths. The distancebetween them (the difference in energy between the hy-perfine components) was calculated [35] using equations(4). Figures 5b and 6b show the width of the indi-vidual Lorentzians (FIG. 7) as the width of the reso-nances changes when at certain magnetic field valuescross-relaxation conditions (equal energy splitting) aremet.

The magnetic field step in the plots of ODMR signalcontrast and width was constrained by our experimen-tal setup. The smallest step was about 0.09 mT (theresolution of the power supply for the electromagnet).

The marked positions in FIG. 5 and FIG. 6 correspondto the transitions from the most populated hyperfinestructure state |mS = 0,mI = +1〉 [38] to some otherempty or much less populated states (see FIG. 10).

The solid, smooth curve (magenta) in FIG. 5 andFIG. 6 shows the result of a Lorentzian fit of the contrastand width data for these cross-relaxation peaks. EachLorentzian peak’s maximum position of the fit curve wasassumed to correspond to the average of the calculatedmagnetic field values for the |mS = 0,mI = +1〉 to the|mS = −1,mI = 0,±1〉 NV center transitions (bold mag-netic field values in the in-figure boxes in FIG. 10).

Five large-amplitude cross-relaxation peaks (peaks A,B, E, H, I in figures 5 and 6) and four smaller peaks (C,D, F, G in figures 5 and 6) are observed.

The analysis of the cross-relaxation peaks in termsof angular momentum projection conservation is compli-cated by the fact that for off-axis P1 centers the naturaldirection of the quantization axis is the direction fromthe P1 center to the NV center with which the cross-relaxation energy exchange is expected – the P1-NV di-rection. The spin projection magnetic quantum numbersmS and mI defined in the P1-NV reference system arenot good quantum numbers any more with respect tothe NV center’s own reference system, which is the NVcenter’s axis direction and coincides in our experimentwith the direction of external magnetic field B. So theangular momentum quantum projection numbers used inthe text should be considered rather as labels to indicatestates.

The group of four overall larger peaks that are sym-metric to the central one correspond to on-axis P1 cross-relaxation transitions (peaks A and I) and off-axis P1cross-relaxation transitions (peaks B and H) that con-

9

serve nuclear spin projection in P1 during the cross-relaxation processes (see FIG. 10). Peaks B and H inthis group of four peaks have a larger amplitude thanpeaks A and I. This amplitude difference is plausible be-cause there is a three times higher probability to havean off-axis P1 center in the vicinity of a particular NVcenter than to have an on-axis P1 center.

Two smallest peaks – C and G correspond to P1 on-axis cross-relaxation transitions in which the nuclear spinprojection of the P1 center is not conserved. Slightlymore intense peaks correspond to similar cross-relaxationtransitions with a change in nuclear spin in the off-axisP1 centers, which are three times more probable.

Looking at the balance of the angular momentum pro-jections including the electron spin as well as the nuclearspin of both partners – the NV center and the P1 center,it is seen from the legend in FIG. 10 that this projectionis not conserved. There are several reasons why spin pro-jection nonconserving transitions in cross-relaxation arepossible. Fist, the phonons and the crystal lattice itselfcan take or provide some of the angular momentum [30].Second, our experiment involves MW photons, which canprovide angular momentum.

Coming back to the cross-relaxation signals, as waspointed out in [16, 17], an intense peak appears aroundthe magnetic field strength values of 51.2 mT – the cen-tral peak E (see FIG. 5 and FIG. 6). This peak corre-sponds simultaneously to a number of on-axis and off-axistransitions (FIG. 10), which accounts for its rather largeamplitude.

VII. CONCLUSIONS

In this study we have demonstrated that ODMR mea-surements on NV centers in diamond crystals can be usedsuccessfully to study cross-relaxation processes betweenNV centers and P1 centers. To interpret the signals pro-duced by these resonances we accounted for the hyperfinestructure in both interaction partners, NV centers and P1centers. In addition, we accounted for the nuclear spinpolarization in the NV centers, which occurs during theprocess of optical polarization of the electron spin of theNV center in the magnetic field due to strong hyperfinelevel mixing. Using these tools we were able to analyzein detail the cross-relaxation resonance peak positions.

Our measurements confirm the presence of five maincross-relaxation transitions (see peaks A, B, E, H, I infigures 5 and 6). This result is in a qualitative agreementwith previous studies [17, 42–44]. However, in contrast tothese cited studies, in our experiment the probabilities oftransitions that in [16] are classified as “disallowed” (tran-sitions C, D, F, and G) are observed as being rather high.The reason for this difference is that our experimentswere performed under continuous MW radiation, whoseabsorbed photons continuously provide angular momen-tum to enable these otherwise “disallowed” transitions.

Cross-relaxation occurs when the energy splitting be-

tween a pair of hyperfine magnetic sublevels in an NVcenter coincides with a hyperfine magnetic sublevel split-ting in substitutional nitrogen. Under these conditionsthe relaxation rate in the NV center effectively increases,ODMR signal contrast decreases, and resonances becomebroadened.

This interpretation of the signal origin is supported bythe experimental evidence. When ODMR signals are ob-served between the hyperfine levels of the electron spintransition |mS = 0〉 −→ |mS = −1〉 (see FIG. 5), thechange in the signal amplitude at the resonance point isaround two times larger than when the resonance is ob-served on the |mS = 0〉 −→ |mS = +1〉 transition (seeFIG. 6). As can be seen from FIG. 1, when we observethe |mS = 0〉 −→ |mS = −1〉 resonances, both levels ofthe transition in the NV center interact resonantly withthe transition in the substitutional nitrogen. On the con-trary, the |mS = 0〉 −→ |mS = +1〉 ODMR transitiondoes not interact resonantly with any particular transi-tion in the substitutional nitrogen. Only the lower level|mS = 0〉 of the NV transition at some particular mag-netic field strength is influenced by the cross-relaxationprocess. As a result, in contrast to the former case, in thisparticular case, only one level of the transition is broad-ened due to cross-relaxation, and as a result the signalcontrast is diminished less effectively and the observedsignal broadening is weaker.

As shown in the model calculation above (Subsec-tion VIB), this difference in effective relaxation ratesis further enhanced by the fact that in the case of|mS = 0〉 −→ |mS = −1〉 resonances, the MW field un-der resonance conditions effectively interacts with bothpartners, the NV center and the substitutional nitrogen,leading to interaction dynamics (see FIG. 9) that can beunderstood as a further increase in the effective relax-ation rate. This further enhancement is not present forthe |mS = 0〉 −→ |mS = +1〉 transition when the MWfield is not in resonance with the transition in the P1center.

The results presented in this paper serve to developnew ways of using NV centers in electron and nuclearmagnetic resonance methods, as well as for studies ofnuclear spin hyperpolarization with NV centers in dia-mond. Furthermore, these results can be important forcross-relaxation studies involving other point-like defectsin a diamond crystal, optimization of experimental condi-tions for microwave-free magnetometry, and microwave-free nuclear-magnetic-resonance probes.

VIII. ACKNOWLEDGEMENTS

We gratefully acknowledge the financial supportfrom the Base/Performance Funding Project Nr.ZD2010/AZ27, AAP2015/B013 of the University ofLatvia. A. Berzins acknowledges support from the Post-Doc Latvia Project Nr. 1.1.1.2/VIAA/1/16/024 "Two-way research of thin-films and NV centres in diamond

10

FIG. 10: ODMR signal width as a function of the applied external magnetic field. The red points are calculated from theexperimental data for the |mS = 0,mI〉 −→ |mS = −1,mI〉 NV center transitions. The tables around the graph show the meanmagnetic field values (bold) of three possible transitions for an NV center in combination with three transitions for the P1center (the same transition at slightly different magnetic field values). The NV center and P1 center transitions are written inthe |mS ,mI〉 bases. Cells and numbers marked in green represent transitions noted to have the total spin projection (magneticquantum number) change’s absolute value |∆m| = |∆mSNV + ∆mINV + ∆mSP1 + ∆mIP1 | less than or equal to 2 (this is trueonly when the projections are along the same axis, otherwise this can be taken only as a classification notation). The off-axistransitions are marked red in the tables.

11

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